Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,6)(4,7)(9,10,13,15)(11,14,16,12)(18,22)(23,24);(1,2,5,6)(3,7,8,4)(9,11,13,16)(10,14,15,12)(17,18,21,22)(19,23,20,24);(2,6)(3,8)(10,15)(11,16);(9,11,13,16)(10,14,15,12);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19,21,20)(18,23,22,24);(1,4,5,7)(2,3,6,8)(9,12,13,14)(10,16,15,11)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239241, #72239255, #72312133, … |
| maximal normal subgroup of | #72239241, #72239255, #72312133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618349);